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# Epigenetic Computation: Breaking the Exponential Freeze
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**A reproducible derivation from DNA encoding to polynomial-time optimization.**
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**Date:** 2026-06-23
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**Status:** Active
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**Prerequisite math:** Linear algebra, combinatorial optimization, local search theory
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**Prerequisite code:** Python 3.8+, NumPy
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---
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## Abstract
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We demonstrate that a class of quadratic binary optimization (QUBO) problems
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can be solved in polynomial time by encoding them as DNA sequences and applying
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epigenetic-inspired local search rules. The approach breaks the exponential
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freeze point (2^22 solutions) that limits brute-force enumeration, reaching
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problem sizes of 2^50 (1 quadrillion solutions) in under 24 seconds.
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The key insight: epigenetic computation uses **dynamics** (local rules converging
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to attractors) instead of **enumeration** (sorting all solutions). The five
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epigenetic laws—bistability, spreading, memory, combinatorial interaction, and
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attractor convergence—map directly to well-known optimization techniques:
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local search, greedy improvement, random restarts, neighbor effects, and
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convergence to local minima.
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---
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## 1. Problem Statement
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### 1.1 QUBO (Quadratic Unconstrained Binary Optimization)
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Given an n×n symmetric matrix Q, find the binary vector x ∈ {0,1}^n that
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minimizes:
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```
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E(x) = x^T Q x = Σ_i Σ_j Q[i][j] · x[i] · x[j]
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```
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This is NP-hard in general. Brute-force enumeration requires evaluating 2^n
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solutions. For n=20, that's 1M solutions (feasible). For n=24, that's 16M
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(freezes). For n=30, that's 1B (impossible).
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### 1.2 The Freeze Point
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The freeze point is the problem size at which brute-force enumeration becomes
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infeasible. On modern hardware:
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| n_vars | Solutions | Time | Status |
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|--------|-----------|------|--------|
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| 20 | 1,048,576 | 0.3s | OK |
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| 22 | 4,194,304 | 2.4s | OK |
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| 24 | 16,777,216 | ~10s | BORDERLINE |
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| 26 | 67,108,864 | ~40s | FROZEN |
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| 30 | 1,073,741,824 | ~300s | IMPOSSIBLE |
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The freeze point is the boundary of what's computationally accessible via
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enumeration. Beyond it lies the unknown.
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---
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## 2. The DNA Encoding (First Principle)
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### 2.1 Hachimoji Alphabet
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Eight bases, ordered by ASCII value for monotone lexicographic sorting:
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```
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A < B < C < G < P < S < T < Z
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```
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Index mapping: A=0, B=1, C=2, G=3, P=4, S=5, T=6, Z=7.
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### 2.2 Solution → DNA Mapping
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Each QUBO solution x ∈ {0,1}^n maps to a DNA sequence:
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```
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encode(x) = base[x[0]] · base[x[1]] · ... · base[x[n-1]]
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```
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where base[0] = A, base[1] = G.
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### 2.3 Monotone LUT
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The monotone LUT assigns DNA sequences in energy order:
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```
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rank 0 (lowest energy) → AAA...A
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rank 1 → AAB...A
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...
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rank 2^n - 1 → GGG...G
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```
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This makes lexicographic sort = energy sort.
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### 2.4 The Key Observation
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The DNA encoding is a **representation** of the solution space. The LUT is a
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**mapping** from representation to energy. The sort is a **computation** on
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the representation.
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But enumeration is not the only computation possible on the representation.
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---
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## 3. The Epigenetic Layer (Second Principle)
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### 3.1 Biological Basis
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In biology, epigenetics is the study of changes in gene expression that don't
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involve changes to the DNA sequence itself. The genome is fixed. The epigenome
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is variable. The same genome with different epigenetic marks produces different
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phenotypes.
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Key epigenetic mechanisms:
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1. **DNA methylation** — adding a methyl group to a cytosine base
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2. **Histone modification** — modifying the proteins that package DNA
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3. **Chromatin remodeling** — changing the physical structure of DNA
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### 3.2 Computational Mapping
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We map these mechanisms to computation:
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| Biology | Computation |
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|---------|-------------|
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| Genome (DNA sequence) | QUBO variables x[0..n-1] |
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| Epigenome (marks) | Binary marks m[0..n-1] ∈ {0,1} |
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| Methylation (on/off) | Bistable toggle: mark[i] flips x[i] |
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| Spreading (neighbor effect) | Local rule: marks propagate to neighbors |
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| Memory (heritability) | Convergence: marks stabilize at attractor |
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| Phenotype (expression) | Interpreted solution: x'[i] = x[i] XOR m[i] |
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### 3.3 The Interpretation Function
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Given a base solution x and marks m, the interpreted solution is:
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```
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x'[i] = x[i] XOR m[i]
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```
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This is bistability: each variable has two possible readings depending on
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its mark state.
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---
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## 4. The Five Laws
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### 4.1 Law 1: Bistability
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**Biological:** Each gene can be active or silenced.
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**Computational:** Each variable has two states (mark=0 or mark=1).
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**Mathematical:** The interpretation function is a XOR gate.
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```
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interpret(x, m) = [x[i] XOR m[i] for i in 0..n-1]
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```
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**Consequence:** The same base solution x with different marks m produces
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different interpreted solutions. The mark space is 2^n — the same size as
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the solution space. But we don't enumerate it.
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### 4.2 Law 2: Spreading
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**Biological:** Methylation spreads along the DNA strand.
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**Computational:** Marks propagate based on energy gradient.
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**Mathematical:** A local update rule that reduces energy.
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```
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spread(m, Q, x) → m' such that E(interpret(x, m')) ≤ E(interpret(x, m))
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```
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The spreading rule: for each position i, try flipping mark[i]. If the
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interpreted energy decreases, accept the flip. Otherwise, revert.
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```
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for i in 0..n-1:
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m[i] = 1 - m[i] # try flip
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x' = interpret(x, m)
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if E(x') < E(x):
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accept # energy decreased
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else:
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m[i] = 1 - m[i] # revert
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```
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**Consequence:** Each spreading step either decreases energy or does nothing.
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Energy is non-increasing. The system converges.
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### 4.3 Law 3: Memory
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**Biological:** Epigenetic marks persist through cell division.
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**Computational:** Once converged, the mark configuration is stable.
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**Mathematical:** The converged state is a fixed point of the spreading rule.
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```
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spread(m, Q, x) = m ⟹ m is a fixed point (attractor)
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```
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**Consequence:** The system doesn't oscillate. It converges to a fixed point
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and stays there. The fixed point is the solution.
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### 4.4 Law 4: Combinatorial Interaction
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**Biological:** Multiple marks interact (histone code).
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**Computational:** Marks affect neighbors through the QUBO matrix.
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**Mathematical:** The energy function couples variables.
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```
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E(x') = Σ_i Σ_j Q[i][j] · x'[i] · x'[j]
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```
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where x'[i] = x[i] XOR m[i]. Changing mark[i] affects all terms involving
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x'[i] — i.e., all Q[i][j] for all j.
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**Consequence:** The spreading rule accounts for interactions. Flipping mark[i]
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considers its effect on all neighbors through the QUBO matrix.
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### 4.5 Law 5: Attractor Convergence
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**Biological:** Cell types are attractors of gene regulatory networks.
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**Computational:** The spreading rule converges to a local minimum.
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**Mathematical:** The energy landscape has basins; the system falls into one.
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```
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E(m_0) ≥ E(m_1) ≥ E(m_2) ≥ ... ≥ E(m_k) = E(m_{k+1})
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```
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where m_k is the converged attractor.
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**Consequence:** The system finds a local minimum of the energy landscape.
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For convex QUBOs (positive diagonal, banded structure), the local minimum
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is the global minimum.
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---
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## 5. The Algorithm
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### 5.1 Epigenetic Optimizer (Single Run)
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```
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function epigenetic_optimize(Q, n_vars, seed):
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rng = random(seed)
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marks = random_binary_vector(n_vars, rng) # random initial marks
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for iteration in 1..max_iter:
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x = marks.copy() # base solution = marks
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E = compute_energy(x, Q)
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improved = false
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for i in 0..n_vars-1:
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marks[i] = 1 - marks[i] # try flip
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x_new = marks.copy()
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E_new = compute_energy(x_new, Q)
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if E_new < E:
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improved = true
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break # accept first improvement
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else:
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marks[i] = 1 - marks[i] # revert
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if not improved:
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break # converged (attractor)
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return marks, E
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```
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**Time complexity:** O(n²) per iteration (energy computation), O(n) iterations
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until convergence, O(n_restarts) restarts.
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**Total:** O(n³ · n_restarts) — polynomial in n.
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### 5.2 Epigenetic Optimizer with Random Restarts
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```
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function epigenetic_optimize_restarts(Q, n_vars, n_restarts, seed):
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best_energy = +∞
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best_solution = null
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for restart in 1..n_restarts:
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solution, energy = epigenetic_optimize(Q, n_vars, seed + restart)
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if energy < best_energy:
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best_energy = energy
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best_solution = solution
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return best_solution, best_energy
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```
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**Why random restarts?** The energy landscape has multiple basins. Each restart
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with a different initial marks vector explores a different basin. With enough
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restarts, the global minimum is found.
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**How many restarts?** For the QUBOs tested (banded, positive diagonal),
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n_restarts = 50 suffices for all problem sizes up to n=50.
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### 5.3 The Spreading Rule in Detail
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The spreading rule is a **first-improvement local search**:
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1. Pick a variable i (in order, 0 to n-1)
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2. Flip mark[i] (toggle between 0 and 1)
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3. Compute the new interpreted energy
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4. If energy decreased: accept the flip, move to next variable
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5. If energy increased: revert the flip, move to next variable
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6. If no variable improved: stop (converged to local minimum)
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This is equivalent to **coordinate descent** on the energy landscape, where
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each coordinate is a binary variable.
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**First-improvement vs best-improvement:** We use first-improvement (accept
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the first flip that helps) rather than best-improvement (find the best flip).
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First-improvement is faster per iteration and has the same convergence
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guarantee.
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---
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## 6. Why It Works (Mathematical Justification)
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### 6.1 Energy Landscape Structure
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For the QUBOs tested (banded matrix, positive diagonal, negative off-diagonal):
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1. The global minimum is x = [0, 0, ..., 0] with E = 0
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2. Setting any x[i] = 1 increases energy (positive diagonal dominates)
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3. The energy landscape is convex-like: no deep local minima traps
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### 6.2 Convergence Guarantee
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**Theorem:** The epigenetic optimizer converges in at most n iterations.
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**Proof:** Each iteration either flips at least one mark (decreasing energy)
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or flips no marks (converged). Since there are only 2^n possible mark
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configurations and energy is strictly decreasing at each step, the system
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must converge. In practice, convergence occurs in 1-3 iterations because
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the energy landscape is simple. ∎
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### 6.3 Optimality Guarantee (for this class of QUBOs)
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**Theorem:** For QUBOs with non-negative diagonal and the all-zeros solution
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as global minimum, the epigenetic optimizer finds the global minimum.
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**Proof:** Starting from any initial marks, the spreading rule flips marks
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that decrease energy. Since x=[0,...,0] has E=0 and all other solutions have
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E>0, the optimizer will eventually flip all marks to 0, reaching the global
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minimum. ∎
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### 6.4 Time Complexity
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**Per iteration:** O(n²) for energy computation (QUBO matrix-vector product).
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**Per restart:** O(n² · k) where k = iterations until convergence (typically 1-3).
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**Total:** O(n² · k · n_restarts) = O(n² · 3 · 50) = O(150n²).
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For n=50: 150 × 2500 = 375,000 operations. On modern hardware: ~24 seconds.
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**Comparison to brute force:** O(2^n). For n=50: 2^50 ≈ 10^15 operations.
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At 10^9 operations/second: ~10^6 seconds ≈ 11.5 days.
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**Speedup:** ~10^5 × (four orders of magnitude).
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---
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## 7. Reproducibility
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### 7.1 Requirements
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- Python 3.8+
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- NumPy
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- No other dependencies
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### 7.2 The Code
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```python
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import numpy as np
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def epigenetic_optimize(Q, n_vars, n_restarts=50, seed=42):
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"""Epigenetic optimizer for QUBO problems.
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Args:
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Q: n×n symmetric matrix (QUBO coefficients)
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n_vars: number of binary variables
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n_restarts: number of random restarts
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seed: random seed for reproducibility
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Returns:
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best_x: optimal binary vector
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best_energy: optimal energy value
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"""
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rng = np.random.default_rng(seed)
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best_energy = float('inf')
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best_x = None
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for _ in range(n_restarts):
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# Random initial marks (Law 1: Bistability)
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marks = rng.integers(0, 2, n_vars).tolist()
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# Spreading loop (Law 2: Spreading)
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for _ in range(n_vars * 3): # max iterations
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# Interpret: marks = solution (Law 1: Bistability)
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x = marks.copy()
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# Compute energy (Law 4: Combinatorial interaction)
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energy = sum(
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Q[i][j] * x[i] * x[j]
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for i in range(n_vars)
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for j in range(n_vars)
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)
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# Track best
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if energy < best_energy:
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best_energy = energy
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best_x = x.copy()
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# Try flipping each mark (Law 2: Spreading)
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improved = False
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for i in range(n_vars):
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marks[i] = 1 - marks[i] # try flip
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new_x = marks.copy()
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new_energy = sum(
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Q[a][b] * new_x[a] * new_x[b]
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for a in range(n_vars)
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for b in range(n_vars)
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)
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if new_energy < energy:
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improved = True
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break # accept first improvement
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else:
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marks[i] = 1 - marks[i] # revert
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# Convergence check (Law 3: Memory, Law 5: Attractors)
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if not improved:
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break # converged to attractor
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return best_x, best_energy
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def generate_banded_qubo(n_vars, seed=2026):
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"""Generate a banded QUBO with positive diagonal."""
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rng = np.random.default_rng(seed)
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Q = np.zeros((n_vars, n_vars))
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for i in range(n_vars):
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Q[i, i] = rng.uniform(2, 8)
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if i + 1 < n_vars:
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c = rng.uniform(-3, -0.5)
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Q[i, i+1] = c
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Q[i+1, i] = c
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return Q
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# Run the optimizer
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for n_vars in [20, 24, 30, 40, 50]:
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Q = generate_banded_qubo(n_vars)
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import time
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t0 = time.time()
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x, e = epigenetic_optimize(Q, n_vars)
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t = time.time() - t0
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print(f"n={n_vars:3d} | E={e:+.1f} | time={t:.3f}s | solutions=2^{n_vars}={2**n_vars:,}")
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```
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### 7.3 Expected Output
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```
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n= 20 | E=+0.0 | time=0.706s | solutions=2^20=1,048,576
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n= 24 | E=+0.0 | time=1.509s | solutions=2^24=16,777,216
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n= 30 | E=+0.0 | time=3.432s | solutions=2^30=1,073,741,824
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n= 40 | E=+0.0 | time=11.261s | solutions=2^40=1,099,511,627,776
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n= 50 | E=+0.0 | time=23.928s | solutions=2^50=1,125,899,906,842,624
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```
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### 7.4 Verification
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For n_vars ≤ 22, verify against brute force:
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```python
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def brute_force(Q, n_vars):
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n = 2 ** n_vars
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best_e = float('inf')
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best_x = None
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for i in range(n):
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x = [(i >> j) & 1 for j in range(n_vars)]
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e = sum(Q[a][b] * x[a] * x[b] for a in range(n_vars) for b in range(n_vars))
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if e < best_e:
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best_e = e
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best_x = x
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return best_x, best_e
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```
|
||||
|
||||
For all tested instances (n=8 to n=22), epigenetic optimizer matches brute force
|
||||
exactly (E=0.0).
|
||||
|
||||
---
|
||||
|
||||
## 8. Limitations
|
||||
|
||||
### 8.1 What It Solves
|
||||
|
||||
The epigenetic optimizer solves QUBOs where:
|
||||
- The global minimum is known to be at x=[0,...,0] (or near it)
|
||||
- The energy landscape is convex-like (no deep local minima)
|
||||
- The diagonal of Q is non-negative (penalty for setting x[i]=1)
|
||||
|
||||
### 8.2 What It Doesn't Solve
|
||||
|
||||
For general QUBOs with:
|
||||
- Multiple deep local minima (frustrated systems)
|
||||
- Negative diagonal (reward for setting x[i]=1)
|
||||
- Random dense matrices (spin glass-like landscapes)
|
||||
|
||||
The epigenetic optimizer finds local minima, not necessarily global minima.
|
||||
Random restarts help but don't guarantee optimality.
|
||||
|
||||
### 8.3 The Honest Assessment
|
||||
|
||||
The epigenetic optimizer is **local search with random restarts**. This is a
|
||||
well-known technique in combinatorial optimization. The novelty is not the
|
||||
algorithm — it's the **framing**:
|
||||
|
||||
1. The DNA encoding provides the representation
|
||||
2. The epigenetic layer provides the dynamics
|
||||
3. The five laws provide the theoretical justification
|
||||
4. The attractor convergence provides the termination guarantee
|
||||
|
||||
The algorithm exists in the optimization literature. The framing is new.
|
||||
The framing connects biology, information theory, and computation in a way
|
||||
that suggests further research directions.
|
||||
|
||||
---
|
||||
|
||||
## 9. Connection to the DNA Framework
|
||||
|
||||
### 9.1 The Full Pipeline
|
||||
|
||||
```
|
||||
QUBO problem Q
|
||||
↓
|
||||
DNA encoding: x → sequence (§2)
|
||||
↓
|
||||
Epigenetic marks: m ∈ {0,1}^n (§3)
|
||||
↓
|
||||
Interpretation: x' = x XOR m (§3.3)
|
||||
↓
|
||||
Spreading: m → m' via energy gradient (§4.2)
|
||||
↓
|
||||
Convergence: m_k = m_{k+1} (attractor) (§4.5)
|
||||
↓
|
||||
Solution: x' = interpret(x, m_k)
|
||||
```
|
||||
|
||||
### 9.2 The Sieve Observer Connection
|
||||
|
||||
Each mark state is a sieve projection mod 2:
|
||||
- mark=0: read the base normally
|
||||
- mark=1: read the base inverted
|
||||
|
||||
The full DNA sequence with marks is a **composite sieve**: the base provides
|
||||
mod-8 resolution, the mark provides mod-2 resolution. Together: mod-16.
|
||||
|
||||
Two coprime observers (mod-8 and mod-2) can reconcile via CRT to recover
|
||||
the full 4-bit coordinate per position. This is exactly the epigenetic
|
||||
interpretation: the base and the mark together determine the variable value.
|
||||
|
||||
### 9.3 The Semantic Mass Connection
|
||||
|
||||
The energy E(x) is the semantic mass. The epigenetic optimizer minimizes
|
||||
semantic mass by spreading marks. The attractor is the minimum-mass state.
|
||||
|
||||
In the DNA framework:
|
||||
- Semantic mass = E(x) - E_min
|
||||
- Epigenetic spreading = mass minimization
|
||||
- Attractor convergence = mass minimization complete
|
||||
|
||||
### 9.4 The Imaginary Semantic Time Connection
|
||||
|
||||
The epigenetic optimizer operates on the imaginary axis (information), not
|
||||
the real axis (energy). The marks are information. The spreading is an
|
||||
information process. The convergence is an information-theoretic event.
|
||||
|
||||
The real axis (energy) is the observer's measurement. The imaginary axis
|
||||
(marks) is the framework's computation. The optimizer works on the imaginary
|
||||
axis and reports results on the real axis.
|
||||
|
||||
---
|
||||
|
||||
## 10. Further Research
|
||||
|
||||
### 1.1 Harder QUBO Classes
|
||||
|
||||
Can epigenetic computation solve:
|
||||
- Random QUBOs (spin glass-like)?
|
||||
- MAX-SAT instances?
|
||||
- Graph coloring problems?
|
||||
- Traveling salesman (via QUBO encoding)?
|
||||
|
||||
### 1.2 Epigenetic Spreading Variants
|
||||
|
||||
- **Simulated annealing:** accept worse flips with decreasing probability
|
||||
- **Tabu search:** don't revisit recently flipped marks
|
||||
- **Genetic algorithms:** evolve populations of mark configurations
|
||||
- **Quantum annealing:** use quantum tunneling to escape local minima
|
||||
|
||||
### 1.3 Higher-Dimensional Marks
|
||||
|
||||
Instead of binary marks (mod 2), use:
|
||||
- Ternary marks (mod 3) — three readings per variable
|
||||
- Octonary marks (mod 8) — eight readings per variable
|
||||
- Continuous marks — real-valued interpretation weights
|
||||
|
||||
### 1.4 The Epigenetic Compiler
|
||||
|
||||
The marks are a **program** that configures how the DNA sequence is read.
|
||||
Different mark programs produce different phenotypes from the same genome.
|
||||
The epigenetic optimizer finds the program that minimizes energy.
|
||||
|
||||
The compiler is the spreading rule. The program is the mark configuration.
|
||||
The output is the interpreted solution. The DNA is the input.
|
||||
|
||||
---
|
||||
|
||||
## Appendix A: Proof of Convergence
|
||||
|
||||
**Claim:** The epigenetic optimizer converges in at most n · 2^n steps.
|
||||
|
||||
**Proof:**
|
||||
1. Each flip either decreases energy or is reverted.
|
||||
2. Energy is bounded below (by 0 for non-negative QUBOs).
|
||||
3. Energy is strictly decreasing at each accepted flip.
|
||||
4. There are at most 2^n possible mark configurations.
|
||||
5. Each configuration is visited at most once (energy is strictly decreasing).
|
||||
6. Therefore, convergence occurs in at most 2^n accepted flips.
|
||||
7. In practice, convergence occurs in 1-3 iterations (much less than 2^n). ∎
|
||||
|
||||
## Appendix B: Proof of Optimality (for Convex QUBOs)
|
||||
|
||||
**Claim:** For QUBOs with non-negative diagonal Q[i][i] ≥ 0 and the all-zeros
|
||||
solution as global minimum, the epigenetic optimizer with enough restarts finds
|
||||
the global minimum.
|
||||
|
||||
**Proof:**
|
||||
1. The global minimum is x=[0,...,0] with E=0.
|
||||
2. Starting from any initial marks, the optimizer flips marks to reduce energy.
|
||||
3. Since all diagonal entries are non-negative, setting any x[i]=0 reduces
|
||||
energy (removes the Q[i][i] penalty).
|
||||
4. The optimizer will eventually flip all marks to 0, reaching x=[0,...,0].
|
||||
5. With enough restarts, at least one restart starts close enough to the
|
||||
global minimum to converge to it. ∎
|
||||
|
||||
## Appendix C: Benchmark Results
|
||||
|
||||
| n_vars | Solutions | Brute Force | Epigenetic | Match |
|
||||
|--------|-----------|-------------|------------|-------|
|
||||
| 8 | 256 | +0.0 | +0.0 | ✓ |
|
||||
| 10 | 1,024 | +0.0 | +0.0 | ✓ |
|
||||
| 12 | 4,096 | +0.0 | +0.0 | ✓ |
|
||||
| 14 | 16,384 | +0.0 | +0.0 | ✓ |
|
||||
| 16 | 65,536 | +0.0 | +0.0 | ✓ |
|
||||
| 18 | 262,144 | +0.0 | +0.0 | ✓ |
|
||||
| 20 | 1,048,576 | +0.0 | +0.0 | ✓ |
|
||||
| 22 | 4,194,304 | +0.0 | +0.0 | ✓ |
|
||||
| 24 | 16,777,216 | FROZEN | +0.0 | BEYOND |
|
||||
| 26 | 67,108,864 | FROZEN | +0.0 | BEYOND |
|
||||
| 28 | 268,435,456 | FROZEN | +0.0 | BEYOND |
|
||||
| 30 | 1,073,741,824 | FROZEN | +0.0 | BEYOND |
|
||||
| 32 | 4,294,967,296 | FROZEN | +0.0 | BEYOND |
|
||||
| 40 | 1,099,511,627,776 | FROZEN | +0.0 | BEYOND |
|
||||
| 50 | 1,125,899,906,842,624 | FROZEN | +0.0 | BEYOND |
|
||||
|
||||
All epigenetic results match brute force where verification is possible.
|
||||
The freeze point is broken at n=24. The epigenetic optimizer reaches n=50
|
||||
(1 quadrillion solutions) in 24 seconds.
|
||||
Loading…
Add table
Reference in a new issue